Zeros and Poles of Z-Transform

All z-transforms in the above examples are rational, i.e., they can be written as a ratio of polynomials of variable s in the general form

X(z)={N(z)}/{D(z)}={k=1}^M (z-z_{0_k})}/{k=1}^N (z-z_{p_k})}

where N(z) is the numerator polynomial of order M with roots z{0_k}, (k=1,2, \cdots, M), and D(z) is the denominator polynomial of order N with roots z_{p_k}, (k=1,2, \cdots, N). In general, we assume the order of the numerator polynomial is lower than that of the denominator polynomial, i.e., M < true="" each="" z0="" zero="" any="" complex="" value="" zp="" for="" which="" vert="" z="z0=H(z0)=0" is="" a="" pole="" numerator="" zeros="" and="" all="" roots="" denominator="" polynomial="" are="" poles="" if="" exceeds="" the="" order="" of="" n="">M), then H(\infty)=0, i.e., there is a zero at infinity. On the other hand, if the order of N(z) exceeds that of D(z) (i.e., M>N), then H(\infty)=\infty, i.e, there is a pole at infinity. On the z-plane zeros and poles can be indicated by o and x respectively. Most essential behavior properties of an LTI system can be obtained graphically from the ROC and the zeros and poles of its transfer function H(z) on the z-plane.